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OBJECTIVE Based on cybernetics, a large system can be divided into subsystems, and the stability of each can determine the overall properties of the system. However, this stability analysis perspective has not yet been employed in electrocardiogram (ECG) signals. This is the first study to attempt to evaluate whether the stability of decomposed ECG subsystems can be analyzed in order to effectively investigate the overall performance of ECG signals, and aid in disease diagnosis. METHODS We used seven different cardiac pathologies (myocardial infarction, cardiomyopathy, bundle branch block, dysrhythmia, hypertrophy, myocarditis, and valvular heart disease) to illustrate our method. Dynamic mode decomposition (DMD) was first used to decompose ECG signals into dynamic modes (DMs) which can be regarded as ECG subsystems. Then, the features related to the DMs stabilities were extracted, and nine common classifiers were implemented for classification of these pathologies. RESULTS Most features were significant for differentiating the above-mentioned groups (value

In this research, the Shehu transform is coupled with the Adomian decomposition method for obtaining the exact-approximate solution of the plasma fluid physical model, known as the Zakharov-Kuznetsov equation (briefly, ZKE) having a fractional order in the Caputo sense. The Laplace and Sumudu transforms have been refined into the Shehu transform. The action of weakly nonlinear ion acoustic waves in a plasma carrying cold ions and hot isothermal electrons is investigated in this study. Important fractional derivative notions are discussed in the context of Caputo. The Shehu decomposition method (SDM), a robust research methodology, is effectively implemented to generate the solution for the ZKEs. A series of Adomian components converge to the exact solution of the assigned task, demonstrating the solution of the suggested technique. Furthermore, the outcomes of this technique have generated important associations with the precise solutions to the problems being researched. Illustrative examples highlight the validity of the current process. The usefulness of the technique is reinforced via graphical and tabular illustrations as well as statistics theory.

In this paper, we are interested in obtaining an approximate numerical solution of the fractional heat equation where the fractional derivative is in Caputo sense. We also consider the heat equation with a heat source and heat loss. The fractional Laplace-Adomian decomposition method is applied to gain the approximate numerical solutions of these equations. We give the graphical representations of the solutions depending on the order of fractional derivatives. Maximum absolute error between the exact solutions and approximate solutions depending on the fractional-order are given. For the last thing, we draw a comparison between our results and found ones in the literature.

This paper examines the use of Lie group and Lie Algebra theory to construct the geometry of pairwise comparisons matrices. The Hadamard product (also known as coordinatewise, coordinate-wise, elementwise, or element-wise product) is analyzed in the context of inconsistency and inaccuracy by the decomposition method. The two designed components are the approximation and orthogonal components. The decomposition constitutes the theoretical foundation for the multiplicative pairwise comparisons.

This paper studies the issue of dissipativity analysis for singular Markovian jump systems (SMJSs) with time-varying delays via improved state decomposition technique. Toward this aim, based on the decomposed state vectors, an improved mode-dependent augmented Lyapunov–Krasovskii functional (LKF) is established for SMJSs after decomposing SJMSs into subsystems. And the constructed augmented LKF relieves the limitation of state decomposition method by adding some single integral functionals. Then, combining the limited Bessel-Legendre integral inequality and the generalized reciprocally convex combination lemma, a less conservative condition that ensures the SMJSs to be stochastically admissible and dissipative is obtained. Finally, the advantages of the derived criterion are verified by two numerical examples.

In this article, the Laplace decomposition method and Modified Laplace decomposition method have been employed to obtain the exact and approximate solutions of the Klein-Gordon equation with the initial profile. An approximate solution obtained by these methods is in good agreement with the exact solution and shows that these approaches can solve linear and nonlinear problems very effectively and are capable to reduce the size of computational work.

The present study explores the oxygen storage capacity of YBaCo4O7+δ prepared by a glycine-complex decomposition method. We reported for the first time that the YBaCo4O7+δ sample was successfully synthesized at such a low temperature of 800 °C by this method. The YBCO-800 N sample exhibited a faster oxygen absorption/desorption speed than that of high calcination temperature samples, and the time required for complete oxygen storage/release was 5 and 6 min at 360 °C, respectively. Moreover, the superior performance observed for this product in the temperature swing adsorption process makes it a promising candidate in oxygen production technologies. This research demonstrated that the glycine-complex decomposition method is an effective method for improving the oxygen storage property of YBaCo4O7+δ and provides a new insight into designing other novel oxygen storage materials.

В работе представлен метод декомпозиции сложных нелинейных систем с использованием процессного подхода. Показаны основные ограничения и условия применения метода к нелинейным системам различной природы и сложности. В предложенном методе процессы являются объектами управления, что позволяет усилить контроль за ними для получения наилучших результатов и повышения эффективности функционирования и прогнозирования работы нелинейной системы. The paper presents a method of decomposition of nonlinear systems with the ability to self-organization using the process approach. The main limitations and conditions for applying the method to nonlinear systems of various nature and complexity are shown. In the proposed method, the processes are objects of control, which allows you to strengthen control over them to obtain the best results and improve the efficiency of functioning and forecasting the operation of a nonlinear system.

SUMMARYOverconstrained manipulators in lower subspaces with unique motions can be created and analyzed. However, far too little attention has been paid to creating a generic method for overconstrained manipulators kinematic analysis. This study aimed to evaluate a generic methodology for kinematic analysis of overconstrained parallel manipulators with partial subspaces (OPM-PS) using decomposition to parallel manipulators (PMs) in lower subspaces. The theoretical dimensions of the method are depicted, and the use of partial subspace for overconstrained manipulators is portrayed. The methodology for the decomposition method is described and exemplified by designing and evaluating the method to two overconstrained manipulators with 5 degrees of freedom (DoF) and 3 DoF. The inverse kinematic analysis is detailed with position analysis and Jacobian along with the inverse velocity analysis. The workspace analysis for the manipulators using the methodology is elaborated with numerical results. The results of the study show that OPM-PS can be decomposed into PMs with lower subspace numbers. As imaginary joints are being utilized in the proposed methodology, it will create additional data to consider in the design process of the manipulators. Thus, it becomes more beneficial in design scenarios that include workspace as an objective.

Operational risk is one of the common risks in organizations, especially in banks, which has a wide range of errors in individual performance or system failure with process problems. In this paper, the mathematical model of operational risk to calculate the probability of the organization being convinced is presented, which is in the form of a Volterra integro-differenial equations and is solved by the Adomian decomposition method (ADM). Also, the ADM is applied in one dimension as semi-Adomian decomposition method (s-ADM). Comparison of results demonstrates proficiency of the ADM and s-ADMin this model.

In this paper, the Sumudu decomposition method (SDM) is modified and applied to solve systems of singular equations of the Lane-Emden-Fowler type. The proposed method is based on the application of Sumudu transform and Adomian decomposition method. Some illustrative examples are given to demonstrate the efficiency of the proposed technique. The results show that the modified method is simple and effective

In this work, an incorporated form of Sadik transform and Adomian decomposition method which is called the Sadik decomposition method is presented. The method is applied to solve a system of nonlinear fractional Volterra integro-differential equations in the convolution form. To avoid collecting the noise terms that lead the method to fail for seeking the solution, the proposed method is modified by selecting a suitable initial solution. The obtained results are expressed in the explicit form of a power series with easily computable terms. In addition, illustrative examples are shown to demonstrate the effectiveness of the method.

In the present paper, a numerical method is proposed to solve the time fractional Newel-Whitead-Segel equation subject to initial condition. This method is based d on the unification of conformable Sumudu transform (CST) and Adomian decomposition method (ADM), and then it is used to find the analytical solutions of linear-nonlinear fractional PDE’s. The test examples are given for illustration

A novel method of jitter decomposition by 1 Dimension-convolutional neural network (1D-CNN) using jitter histogram points is proposed for decomposing in the time interval error (TIE) of oscilloscope. Unlike the traditional jitter decomposition method based on the dual-Dirac model and Gaussian mixture model (GMM), the proposed jitter decomposition can reduce complexity and improve accuracy. The proposed method consists of a four-layer 1D-convolutional layer and multi-layer perceptron (MLP). Experimental results show that the proposed approach can decompose the total jitter (TJ) into deterministic jitter (DJ) and random jitter (RJ) and is better than the traditional jitter decomposition method, i.e., GMM, Fast Fourier transform and time lag correlation (FFT+TLC). Besides, results show that the proposed method’s performance is better than 2D-PointCNN, PointRNN, CNN, and PointANN.

Link prediction is a key tool for studying the structure and evolution mechanism of complex networks. Recommending new friend relationships through accurate link prediction is one of the important factors in the evolution, development, and popularization of social networks. At present, scholars have proposed many link prediction algorithms based on the similarity of local information and random walks. These algorithms help identify actual missing and false links in various networks. However, the prediction results significantly differ in networks with various structures, and the prediction accuracy is low. This study proposes a method for improving the accuracy of link prediction. Before link prediction, k-shell decomposition method is used to layer the network, and the nodes that are in 1-shell and the nodes that are not linked to the high-shell in the 2-shell are deleted. The experiments on four real network datasets verify the effectiveness of the proposed method.

The construction of smart grid is an important part of improving the utilization rate of electric energy. As an important way for the construction of smart grid, non-intrusive load decomposition methods have been extensively studied. In this type of method, limited by transmission cost and network bandwidth, low-frequency data has been widely used in practical applications. However, the accuracy of device identification in this case faces challenges. Due to the relatively single characteristics of low-frequency data, it is difficult to express the operating status of complex electrical appliances, resulting in low decomposition performance. In this paper, a non-intrusive load is proposed based on household electrical habits by studying the relationship between household electricity consumption habits and load status decomposition method. The Gaussian mixture model and time information are used to model the probability distribution of the electrical appliance state. This probability distribution is then used as the observation probability distribution of the factor hidden Markov model. In such a way, the BH-FHMM model is proposed. Finally, load decomposition is carried out through the load decomposition process of the FHMM model. In order to verify the performance of the proposed method, an experimental comparison is conducted based on the REDD data set. According to the results, a significant improvement in equipment recognition accuracy is obtained.

A method is proposed for decomposing a sum of Gaussian functions to resolve an experimental photoluminescence spectrum into individual Gaussian bands. The decomposition process is based on a single experimental measurement. Here the parameters of a Gaussian (amplitude, location of maximum, and spectral half width) are calculated using three experimental points. The method has been tested in an analysis of the photoluminescence spectra of ZnS:Mn crystals in which the individual bands were revealed by the proposed method as well as by the Alentsev–Fock method and by derivative spectroscopy.

A microservices identification method was proposed by this research paper. The proposed method consists of two parts; the first part is representing the source code of the monolithic application as...

In this paper, we consider the issue of exponential stability analysis for discrete-time quaternion-valued neural networks(DTQVNNs) with leakage delay and discrete time-varying delays. The DTQVNNs are investigated directly rather than through real decomposition method or plural decomposition method. Firstly, the existence and uniqueness of the equilibrium point for DTQVNNs are proposed by using homeomorphic mapping theorem and Cauchy-Schwarz inequality. Then, based on Lyapuinov-Krasovskii functional and matrix inequality, a delay-dependent sufficient condition is provided to guarantee the exponential stability of the equilibrium point for DTQVNNs. Finally, a numerical example is utilized to illustrate the effectiveness of the obtained results.

SummaryThe analysis of gas production from fractured ultralow-permeability (ULP) reservoirs is most often accomplished using numerical simulation, which requires large 3D grids, many inputs, and typically long execution times. We propose a new hybrid analytical/numerical method that reduces the 3D equation of gas flow into either a simple ordinary-differential equation (ODE) in time or a 1D partial-differential equation (PDE) in space and time without compromising the strong nonlinearity of the gas-flow relation, thus vastly decreasing the size of the simulation problem and the execution time.We first expand the concept of pseudopressure of Al-Hussainy et al. (1966) to account for the pressure dependence of permeability and Klinkenberg effects, and we also expand the corresponding gas-flow equation to account for Langmuir sorption. In the proposed hybrid partial transformational decomposition method (TDM) (PTDM), successive finite cosine transforms (FCTs) are applied to the expanded, pseudopressure-based 3D diffusivity equation of gas flow, leading to the elimination of the corresponding physical dimensions. For production under a constant- or time-variable rate (q) regime, three levels of FCTs yield a first-order ODE in time. For production under a constant- or time-variable pressure (pwf) regime, two levels of FCTs lead to a 1D second-order PDE in space and time. The fully implicit numerical solutions for the FCT-based equations in the multitransformed spaces are inverted, providing solutions that are analytical in 2D or 3D and account for the nonlinearity of gas flow.The PTDM solution was coded in a FORTRAN95 program that used the Laplace-transform (LT) analytical solution for the q-problem and a finite-difference method for the pwf problem in their respective multitransformed spaces. Using a 3D stencil (the minimum repeatable element in the horizontal well and hydraulically fractured system), solutions over an extended production time and a substantial pressure drop were obtained for a range of isotropic and anisotropic matrix and fracture properties, constant and time-variable Q and pwf production schemes, combinations of stimulated-reservoir-volume (SRV) and non-SRV subdomains, sorbing and nonsorbing gases of different compositions and at different temperatures, Klinkenberg effects, and the dependence of matrix permeability on porosity. The limits of applicability of PTDM were also explored. The results were compared with the numerical solutions from a widely used, fully implicit 3D simulator that involved a finely discretized (high-definition) 3D domain involving 220,000 elements and show that the PTDM solutions can provide accurate results for long times for large well drawdowns even under challenging conditions.Of the two versions of PTDM, the PTD-1D was by far the better option and its solutions were shown to be in very good agreement with the full numerical solutions, while requiring a fraction of the memory and orders-of-magnitude lower execution times because these solutions require discretization of only the time domain and a single axis (instead of three). The PTD-0D method was slower than PTD-1D (but still much faster than the numerical solution), and although its solutions were accurate for t The PTDM is an entirely new approach to the analysis of gas flow in hydraulically fractured ULP reservoirs. The PTDM solutions preserve the strong nonlinearity of the gas-flow equation and are analytical in 2D or 3D. This being a semianalytical approach, it needs very limited input data and requires computer storage and computational times that are orders-of-magnitude smaller than those in conventional (numerical) simulators because its discretization is limited to time and (possibly) a single spatial dimension.

The paper proposes a novel hybrid method that extends previous work incorporating the fractionally cointegrated vector autoregressive and the permanent-transitory decomposition model. Using the hyb...

This research proposed a novel decomposition method for refactoring monolithic applications into microservices applications using a neural network model (code2vec) for creating code embeddings from the monolithic application source code. As a Result, semantically similar code embeddings are clustered through a hierarchical clustering algorithm to produce microservices candidates to resemble the domain model more efficiently. The quality characteristics of the results were measured using two metrics for measuring cohesion. These metrics were Cohesion at Message Level (CHM) and Cohesion at Domain Level (CHD). Also, four applications were used as test cases with different sizes ranging from small to big applications. The proposed method showed promising results in terms of cohesion when compared to other decomposition methods. The proposed method scored better scores in 5 out of 8 tests compared to other methods. Also, averaged CHD and CHM results were 0.52 and 0.76, respectively, for the proposed method, better results when compared to the other methods.

In this study, the closed form of series solutions of the original and modified nonlinear time-fractional Fornberg-Whitham equations are derived by means of the Laplace decomposition method (LDM). The fractional order derivatives are expressed in the sense of Caputo. For the specific choice of parameters, the obtained solutions are compared with the exact solutions to validate the accuracy of this method. Numerical solutions are represented graphically which illustrate the behavior of the solutions. Further, the computations express that the above method is straightforward, and it desires the smaller size of computation.

Microgrids can be distinguished from traditional power systems based on their ability to perform the islanded operation. This study explores the problem of optimizing the microgrid operation in the context of uncertain islanding events. This occurs when the microgrid is isolated from the main grid and it operates as an independent system. A multistage stochastic optimization model that considers multi-occurrence and multi-period islanding events is proposed to optimize the proactive policy. This is achieved by creating parameters for the maximum number of time periods of the islanded operation in the planning horizon. To solve the resulting large-scale mixed-integer program, an efficient algorithm based on the two-stage Benders’ decomposition method is proposed. Numerical experiments show that the proposed policy significantly reduces the expected operation costs of a microgrid. This is more cost-effective in comparison to the other reactive policies that prepare a certain reserve level and reschedule the resources after an islanding event occurs. This study demonstrates that the proposed decomposition algorithm can efficiently solve large-scale problems that have longer planning horizons or a larger number of time periods for the islanded operation.

The current work is devoted to bring out a detail analysis including qualitative and semi-analytical study of Hepatitis B model under the Caputo- Fabrizio fractional derivative (CFFD). For the required results, fixed point theory is used to establish the conditions for existence and uniqueness of solution to the considered model. On the other hand, for semi analytical solutions, we use decomposition method of Adomian coupled with integral transform of Laplace. Moreover, the concerned solutions are presented via graphs to analyze the dynamics of different compartments of the model.

In this article, exact solutions of some Laplace-type fractional boundary value problems (FBVPs) are investigated via natural decomposition method. The fractional derivatives are described within Caputo operator. The natural decomposition technique is applied for the first time to boundary value problems (BVPs) and found to be an excellent tool to solve the suggested problems. The graphical representation of the exact and derived results is presented to show the reliability of the suggested technique. The present study is mainly concerned with the approximate analytical solutions of some FBVPs. Moreover, the solution graphs have shown that the actual and approximate solutions are very closed to each other. The comparison of the proposed and variational iteration methods is done for integer-order problems. The comparison, support strong relationship between the results of the suggested techniques. The overall analysis and the results obtained have confirmed the effectiveness and the simple procedure of natural decomposition technique for obtaining the solution of BVPs.

The aim of this paper is to present a comprehensive literature review on the application of decomposition methods of time series forecasting in power systems. A comprehensive search is conducted, relevant publications are identified and summarised in terms of the aim of the forecast, decomposition forecasting method used, and a comparison is made with other forecasting techniques. Moreover, publications are analysed by number of publications per year, number of papers by the decomposition method used, and number of publications by area of application of the method in power systems. It is shown that the number of publications per annum grew substantially after 2014 due to studies on the application of decomposition methods in forecasting distributed generation output. Of the methods of forecasting, most publications have used multiplicative decomposition, a lower number has used additive decomposition, with the balance of the papers only using decomposition of the time series as an initial phase towards forecasting using other techniques. The analysis of the papers also shows that decomposition methods have been used in power systems mainly for load, price and distributed generation forecasting. The review contributes to literature by providing an extensive overview of the papers and a repository of references to researchers interested in decomposition forecasting techniques.

Nonlinear initial value problems are somewhat difficult to solve analytically as well as numerically related to linear initial value problems as their variety of natures. Because of this, so many scientists still searching for new methods to solve such nonlinear initial value problems. However there are many methods to solve it. In this article we have discussed about the approximate solution of nonlinear first order ordinary differential equation using ZZ decomposition method. This method is a combination of the natural transform method and Adomian decomposition method.

In this article, the magnetohydrodynamic (MHD) of Jeffery-Hamel flow (JHF) of viscoelastic fluid in converging/diverging channels is investigated, the viscoelastic fluids and when the stationary ri...

Newell Whitehead Segal (NWS) equation has been used in describing many natural phenomena arising in fluid mechanics and hence acquired more attention. Studies in the past gave importance to obtaining numerical or analytical solutions of this kind of equations by employing methods like Modified Homotopy Analysis Transform method (MHATM), Adomian Decomposition method (ADM), Homotopy Analysis Sumudu Transform method (HASTM), Fractional Complex Transform (FCT) coupled with He's polynomials method (FCT-HPM) and Fractional Residual Power Series method (FRPSM). This research aims to demonstrate an efficient analytical method called the Sumudu Decomposition Method (SDM) for the study of analytical and numerical solutions of the NWS of fractional order. The coupling of Adomian Decomposition method with Sumudu transform method simplifies the calculation. From the numerical results obtained, it is evident that SDM is easy to execute and offers accurate results for the NWS equation than with other methods such as FCT-HPM and FRPSM. Therefore, it is easy to apply the coupling of Adomian Decomposition technique with Sumudu transform method, and when applied to nonlinear differential equations of fractional order, it yields accurate results.

Some modified techniques are used in this article in order to have approximate solutions for systems of Volterra integro-differential equations. The suggested techniques are the so called Laplace-Adomian decomposition method and Laplace iterative method. The proposed methods are robust and accurate as can be seen from the given illustrative examples and from the comparison that are made with the exact solution.

Complete decomposition is an essential prerequisite for accurate trace element quantification in geological samples. This work presents a general high-pressure closed digestion approach for rock samples from basic to acidic rocks by inductively coupled plasma mass spectrometry. Using HNO3−HF mixed system, different geological reference materials have been investigated for trace element determination, with acid ratio, decomposition time, digestion temperature, sample mass and reagent amount discussed in detail. In brief, 2.0 mL of HNO3−HF with the ratio of 1 : 1 and a digestion time of 12 h at 185°C in high-pressure sealed bomb are optimal for 50 mg rock sample decomposition. With relative errors under 10%, the analytical results of W-2a, BCR-2, GSP-2, AGV-2 and GSR-1 agree well with certified values. This proposed high-pressure sealed digestion method is characterized with less acid consumption, complete digestion and less damage for digestion process, well meeting the requirements for large sample throughput in geological laboratory.

The decomposition of multi-component micro-Doppler signals overlapping in the time-frequency (T-F) domain is critical and challenging, especially in the case of irregular instantaneous Doppler frequency. The authors propose a novel time-based decomposition method called the short-time variational mode decomposition (STVMD) to analyse the irregular (FM) micro-Doppler signals, and present an optimal model combined with T-F transformation. Then, considering the STVMD may fail to extract the instantaneous frequency (IF) of overlapped components, an improved STVMD algorithm is put forward. Since the dependence of the STVMD algorithm on the initial value, they adopt the Kalman filtering to implement IF tracking and regrouping under the global constraint, further accelerating the convergence of the algorithm. Furthermore, due to the mode aliasing at the intersection point, they adopt a degenerate STVMD model to decompose the signals with known centre frequencies, which can be viewed as a Wiener filter. With the two steps, the improved STVMD algorithm can effectively solve the decomposition of T-F overlapping irregular FM micro-Doppler signals. Compared with the peak ridge technique and the ridge path regrouping and intrinsic chirp component decomposition (RPRG + ICCD), the proposed method shows the effectiveness and adaptability even for irregular FM signals with large T-F spectrum amplitude fluctuation in the low signal-to-noise ratio environment.

In this paper, we present the solution of Black-Scholes fractional partial differential equation with two assets in the Caputo fractional derivative sense by the Aboodh decomposition method. This method is a mixture between new transform called Aboodh transform and the Adomian decomposition method. © 2020. All Rights Reserved.

Optical fiber cables deployed in the last 30 years have been considered to be long-lived components due to the generally high level of reliability of glass and the robustness of cable construction. Like all other components however, optical cables can change with age as a result of their construction or deployment characteristics as well as operational and environmental stress factors. In this article, we apply a time-series decomposition method on span loss data collected during 12 months in four bidirectional spans of a production network in order to detect long-term degradation of the fiber plant. After extracting the trend component, a Mann–Kendall test is applied to the span loss curves and a Sen's slope estimator test is used for determining the magnitude of span loss change. The method allows detecting a 1.3% increase in span loss over the 12-month period of observation in one of the fiber spans. This trend is confirmed using a linear regression model computed according to Theil Sen method applied to additional one-year data. The proposed method can allow the early detection of fiber plant degradation and the proactive planning of fiber replacement.

In this paper, we consider the finite time anti-synchronization (A-SYN) of master-slave coupled quaternion-valued neural networks, where the time-varying delays can be asynchronous and unbounded. Without adopting the general decomposition method, the quaternion-valued state is considered as a whole, which greatly reduces the hassle of further analysis and calculations. The designed controller is delay-free, and the terms with time delay do not need to be bounded globally. Several sufficient conditions for ensuring the finite time A-SYN are obtained under 1-norm and 2-norm respectively. The A-SYN error will be proved to evolve from the initial value to 1 in finite time, and evolve from 1 to 0 also in finite time, hence the finite time A-SYN is proved, which is called two-phases-method. Moreover, adaptive rules for control strengths are also designed to realize the finite time A-SYN. Lastly, a numerical example is presented to demonstrate the correctness and effectiveness of our obtained results.

In this study, the double Laplace Adomian decomposition method and the triple Laplace Adomian decomposition method are employed to solve one- and two-dimensional time-fractional Navier–Stokes problems, respectively. In order to examine the applicability of these methods some examples are provided. The presented results confirm that the proposed methods are very effective in the search of exact and approximate solutions for the problems. Numerical simulation is used to sketch the exact and approximate solution.

In this article, we applied a new technique for solving the time-fractional coupled Korteweg-de Vries (KdV) equation. This method is a combination of the natural transform method with the Adomian decomposition method called the natural decomposition method (NDM). The solutions have been made in a convergent series form. To demonstrate the performances of the technique, two examples are provided.